A combinatorial classification of 2-regular simple modules for Nakayama algebras |
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Authors: | René Marczinzik Martin Rubey Christian Stump |
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Affiliation: | 1. Institute of Algebra and Number Theory, University of Stuttgart, Germany;2. Fakultät für Mathematik und Geoinformation, TU Wien, Austria;3. Fakultät für Mathematik, Ruhr-Universität Bochum, Germany;1. Department of Mathematical Sciences, Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560, Japan;2. Graduate School of Science and Engineering, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560, Japan;1. Department of Applied Mathematics, Ulyanovsk State University, Ulyanovsk 432970, Russia;2. Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, Ed. 8, 90128 Palermo, Italy;1. Department of Mathematics, Northeastern University, Boston, MA 02115, USA;2. Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações, Instituto de Investigação e Formação Avançada, Universidade de Évora, Rua Romão Ramalho, 59, P-7000-671 Évora, Portugal |
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Abstract: | Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a combinatorial classification of 2-regular simple modules for Nakayama algebras and we use this classification to answer several natural questions such as when there is a unique exact structure on the category of finitely generated projective modules for Nakayama algebras. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of global dimension at most two. It turns out that most classes are enumerated by well-known combinatorial sequences, such as Fibonacci, Riordan and Narayana numbers. We first obtain interpretations in terms of the Auslander-Reiten quiver of the algebra using homological algebra, and then apply suitable bijections to relate these to combinatorial statistics on Dyck paths. |
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Keywords: | Nakayama algebras Quiver representation theory Homological algebra Dyck paths Bijective combinatorics Combinatorial statistics |
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